Learning Math: Patterns, Functions, and Algebra
Functions and Algorithms Part A: Doing and Undoing (15 minutes)
Session 3, Part A
There are many situations, both in and outside of mathematics, where the process of doing and undoing helps you organize your activities and figure out how to reverse what you’ve done. In mathematics, it is often important to know how to undo an operation. Here are some examples from everyday life and mathematics:
- School buses pick up children every morning and then drop them off in the same spots every afternoon. Routes are usually organized by a “first on, last off” routine.
- You put on socks and then shoes every morning, and you take off shoes and then socks every night.
- If you added 3 to a number and got 724, you can get your original number back by subtracting 3.
Sometimes you do things that can’t be undone:
- If the cover comes off the hot pepper shaker while you’re sprinkling it on the pizza, there’s not much you can do to undo the process.
- If you mix blue laundry detergent and water, you’d have a hard time separating them back into their original components.
- If you subtracted 10 from a number, then multiplied the result by itself, you wouldn’t be able to find, with certainty, the original number just from undoing the steps.
How can you tell that you wouldn’t be able to definitively find the original number in the numerical rule given above, in which 10 is subtracted from a number and then that number is multiplied by itself?
Tip: See if you can find two different inputs whose outputs are the same. How would that make it impossible to find the original number?
Problem A2: Write and Reflect
Give some examples from teaching, mathematics, or anywhere else where doing and undoing comes into play.
Problem A3: Write and Reflect
Give an example of something you wish you could undo, but the undoing is impossible.
This exercise on “doing and undoing” will be our first look at functions.
Read through the examples of things that can be done and undone, and things that cannot be undone.
Groups: Discuss the examples as a large group. You may want to discuss the number puzzle that cannot be undone. Then, take five minutes to discuss Problems A2 and A3 in pairs.
The inputs 12 and 8 each lead to the output 4. If you only knew that the output was 4, it would be impossible to determine which of 12 and 8 was the correct input.
- Driving directions. Telling someone how to get somewhere usually allows them to figure out how to get back (if there are no one-way streets involved). In particular, to drive back you must take each road the opposite direction, and in reverse order (last road first).
- Packing and unpacking. Especially with commercially shipped packages, it can be difficult to re-pack a box without knowing where things were located before you unpacked it.
- In mathematics, many algebra problems involve “undoing” steps. For example, if 3x = 12, you can find x by undoing the multiplication step.
Lots of things can’t be undone easily, like throwing a water balloon, using gasoline in a car engine, or exploding fireworks.
Session 1 Algebraic Thinking
In this initial session, we will explore algebraic thinking first by developing a definition of what it means to think algebraically, then by using algebraic thinking skills to make sense of different situations.
Session 2 Patterns in Context
Explore the processes of finding, describing, explaining, and predicting using patterns. Topics covered include how to determine if patterns in tables are uniquely described and how to distinguish between closed and recursive descriptions. This session also introduces the idea that there are many different conceptions of what algebra is.
Session 3 Functions and Algorithms
In Session 1, we looked at patterns in pictures, charts, and graphs to determine how different quantities are related. In Session 2, we used patterns and variables to describe relationships in tables and in situations like toothpicks and triangles. This session extends the exploration of relationships to include the concepts of algorithm and function. Note1
Session 4 Proportional Reasoning
Look at direct variation and proportional reasoning. This investigation will help you to differentiate between relative and absolute meanings of "more" and to compare ratios without using common denominator algorithms. Topics include differentiating between additive and multiplicative processes and their effects on scale and proportionality, and interpreting graphs that represent proportional relationships or direct variation.
Session 5 Linear Functions and Slope
Explore linear relationships by looking at lines and slopes. Using computer spreadsheets, examine dynamic dependence and linear relationships and learn to recognize linear relationships expressed in tables, equations, and graphs. Also, explore the role of slope and dependent and independent variables in graphs of linear relationships, and the relationship of rates to slopes and equations.
Session 6 Solving Equations
Look at different strategies for solving equations. Topics include the different meanings attributed to the equal sign and the strengths and limitations of different models for solving equations. Explore the connection between equality and balance, and practice solving equations by balancing, working backwards, and inverting operations.
Session 7 Nonlinear Functions
Continue exploring functions and relationships with two types of non-linear functions: exponential and quadratic functions. This session reveals that exponential functions are expressed in constant ratios between successive outputs and that quadratic functions have constant second differences. Work with graphs of exponential and quadratic functions and explore exponential and quadratic functions in real-life situations.
Session 8 More Nonlinear Functions
Investigate more non-linear functions, focusing on cyclic and reciprocal functions. Become familiar with inverse proportions and cyclic functions, develop an understanding of cyclic functions as repeating outputs, work with graphs, and explore contexts where inverse proportions and cyclic functions arise. Explore situations in which more than one function may fit a particular set of data.
Session 9 Algebraic Structure
Take a closer look at "algebraic structure" by examining the properties and processes of functions. Explore important concepts in the study of algebraic structure, discover new algebraic structures, and solve equations in these new structures.
Session 10 Classroom Case Studies, Grades K-2
Explore how the concepts developed in Patterns, Functions, and Algebra can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This video is for the K-2 grade band.
Session 11 Classroom Case Studies, Grades 3-5
Explore how the concepts developed in Patterns, Functions, and Algebra can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This video is for the 3-5 grade band.
Session 12 Classroom Case Studies, Grades 6-8
Explore how the concepts developed in Patterns, Functions, and Algebra can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This video is for the 6-8 grade band.